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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomatic-deductive proofs are not a posteriori work, a luxury we can marginalize nor are computer-assisted proofs bad mathematics. There is hope for integration! 1

Citations

...ation, visualisation tools, simulation and data–mining. New types of proofs motivated by the experimental “ideology” have appeared. For example, the interactive proof (see Goldwasser, Micali, Rackoff =-=[18]-=-, Blum [5]) or the holographic proof (see Babai [3]). And, of course, these new developments have put the classical idea of axiomatic-deductive proof under siege (see [11] for a detailed discussion). ...

... the deterministic chaos phenomenon, fractals. Wolfram’s extensive computer experiments in theoretical physics paved the way for his discovery of simple programs having extremely complicated behavior =-=[40]-=-. Experimental mathematics—as systematic mathematical experimentation ranging from hypotheses building to assisted proofs and automated proof–checking—will play an increasingly important role and will...

...mple! A moment of reflection shows that the case may not be so simple. For example, what if the “agent” (human or computer) checking a proof for correctness makes a mistake (as pointed out by Lakatos =-=[24]-=-, agents are fallible)? Obviously, another agent has to check that the agent doing the checking did not make any mistake. Some other agent will need to check that agent, and so on. Eventually either t...

...alisation tools, simulation and data–mining. New types of proofs motivated by the experimental “ideology” have appeared. For example, the interactive proof (see Goldwasser, Micali, Rackoff [18], Blum =-=[5]-=-) or the holographic proof (see Babai [3]). And, of course, these new developments have put the classical idea of axiomatic-deductive proof under siege (see [11] for a detailed discussion). Two progra...

...ly’, have created a blend of logical and empirical–experimental arguments which is called “quasi–empirical mathematics” (by Tymoczko [39], Chaitin [13]) or “experimental mathematics” (Borwein, Bailey =-=[8]-=-, Borwein, Bailey, Girgensohn [9]). Mathematicians increasingly use symbolic and numerical computation, visualisation tools, simulation and data–mining. New types of proofs motivated by the experiment...

...Things don’t simply have to make sense to another human being, they must make sense to a computer. And indeed, Knuth compared his TEX compiler (a document of about 500 pages) with Feit and Thompson’s =-=[17]-=- theorem that all simple groups of odd order are cyclic. He lucidly argues that the program might not incorporate as much creativity and “daring” as the proof of the theorem, but they come even when c...

...rudence It is well-known that the doubt appeared in respect to the Hilbert-Bourbaki rigor was caused by Gödel’s incompleteness theorem, 4 see, for instance, Kline’s Mathematics, the Loss of Certainty =-=[21]-=-. It is not by chance that a similar title was used later by Ilya Prigogine in respect to the development of physics. So, arrogance was more and more replaced by prudence. All rigid attitudes, based o...

...mpt to formalise all of mathematics) to optimism (a guarantee that mathematics will go on forever) or simple dismissal (as irrelevant for the practice of mathematics). See more in Barrow [4], Chaitin =-=[12]-=- and Rozenberg and Salomaa [34]. The main pragmatical conclusion seems to be that ‘mathematical knowledge’, whatever this may mean, cannot solely be derived only from some fixed rules. Then, who valid...

...here we had several ‘Gorensteins’, not only one, does not essentially change the situation. How do exceedingly long proofs compare with assisted proofs? In 1996 Robertson, Sanders, Seymour and Thomas =-=[32]-=- offered a simpler proof of the 4CP. They conclude with the following interesting comment (p. 24): We should mention that both our programs use only integer arithmetic, and so we need not be concerned...

...e of any attempt to formalise all of mathematics) to optimism (a guarantee that mathematics will go on forever) or simple dismissal (as irrelevant for the practice of mathematics). See more in Barrow =-=[4]-=-, Chaitin [12] and Rozenberg and Salomaa [34]. The main pragmatical conclusion seems to be that ‘mathematical knowledge’, whatever this may mean, cannot solely be derived only from some fixed rules. T...

...somebody gives you what they claim is a proof, there is a mechanical procedure that will check whether the proof is correct or not, whether it obeys the rules or not. And according to Jaffe and Quinn =-=[20]-=- Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability...

.... In argumentation theory, various ways to argue are discussed, deductive reasoning being only one of them. The literature in this respect goes from classical rhetorics to recent developments such as =-=[28]-=-. People argue by all means. We use suggestions, impressions, emotions, logic, gestures, mimicry, etc. What is the relation between proof in general and proof in mathematics? It seems that the longer ...

...grams, have penetrated massively in mathematics and led to a solution of the four-color problem (4CP), a solution which is still an object of debate and controversy, see Appel and Haken [1], Tymoczko =-=[39]-=-, Swart [37], Marcus [27], and A. Calude [10]. 3 Clearly, any proof, be it mathematical or not, is a very heterogeneous process, where different ingredients are involved in various degrees. The increa...

...verified in the same way as traditional mathematical proofs. We concede, however, that verifyingacomputer program is much more difficult than checking a mathematical proof of the same length. 5 Knuth =-=[22]-=- p. 18 confirms the opinion expressed in the last lines of the previous paragraph: . . . program–writing is substantially more demanding than book–writing. Why is this so? I think the main reason is t...

...cs and led to a solution of the four-color problem (4CP), a solution which is still an object of debate and controversy, see Appel and Haken [1], Tymoczko [39], Swart [37], Marcus [27], and A. Calude =-=[10]-=-. 3 Clearly, any proof, be it mathematical or not, is a very heterogeneous process, where different ingredients are involved in various degrees. The increasing role of empirical-experimental factors m...

...e computer programs, have penetrated massively in mathematics and led to a solution of the four-color problem (4CP), a solution which is still an object of debate and controversy, see Appel and Haken =-=[1]-=-, Tymoczko [39], Swart [37], Marcus [27], and A. Calude [10]. 3 Clearly, any proof, be it mathematical or not, is a very heterogeneous process, where different ingredients are involved in various degr...

...or definitions to proofs and theorems, may completely forget about the possibility and importance of naive guessings20 Cristian S. Calude and Solomon Marcus some from eminent mathematicians as Atiyah =-=[2]-=-: [20] present a sanitized view of mathematics which condemns the subject to an arthritic old age. They see an inexorable increase in standards and are embarrassed by earlier periods of sloppy reasoni...

...ng. New types of proofs motivated by the experimental “ideology” have appeared. For example, the interactive proof (see Goldwasser, Micali, Rackoff [18], Blum [5]) or the holographic proof (see Babai =-=[3]-=-). And, of course, these new developments have put the classical idea of axiomatic-deductive proof under siege (see [11] for a detailed discussion). Two programatic ‘institutions’ are symptomatic for ...

...icians familiar with Perelman’s work expect that it will be difficult to locate any substantial mistakes, cf. Robinson [33]. 7 The conjugate pair rigor-meaning deserves to be reconsidered, cf. Marcus =-=[26]-=-. 8 Knuth’s concept of treating a program as a piece of literature, addressed to human beings rather than to a computer; see [23].sMathematical Proofs at a Crossroad? 25 are supposed to be the same, a...

...penetrated massively in mathematics and led to a solution of the four-color problem (4CP), a solution which is still an object of debate and controversy, see Appel and Haken [1], Tymoczko [39], Swart =-=[37]-=-, Marcus [27], and A. Calude [10]. 3 Clearly, any proof, be it mathematical or not, is a very heterogeneous process, where different ingredients are involved in various degrees. The increasing role of...

...non-formal arguments. ‘Mathematics’ means ‘proof’ and ‘proof’ means ‘formal proof’, is the new slogan. Depuis les Grecs, qui dit Mathématique, dit démonstration is Bourbaki’s slogan, while Mac Lane’s =-=[25]-=- austere doctrine reads If a result has not yet been given valid proof, it isn’t yet mathematics: we should strive to make it such. Here, the proof is conceived according to the standards established ...

...Are these two scenarios incompatible? Not at all. It happens that the second scenario was systematically neglected; but the historical reasons for this mistake will not be discussed here (see more in =-=[29, 30]-=-). Going back to proof, perhaps the most important task of mathematical education is to explain why, in many circumstances, informal statements of problems and informal proofs are not sufficient; then...

... of the finite simple groups required a total of about fifteen thousand pages, spread in five-hundred separate articles belonging to about three-hundred different authors (see Conder [15]). But Serre =-=[31]-=- is still waiting for experts to check the claim by Aschbacher and Smith to have succeeded filling in the gap in the proof of the classification theorem, a gap already discovered in 1980 by Daniel Gor...

...rd proofs based on diagrams which are claimed to commute, arrows which 6 Mathematicians familiar with Perelman’s work expect that it will be difficult to locate any substantial mistakes, cf. Robinson =-=[33]-=-. 7 The conjugate pair rigor-meaning deserves to be reconsidered, cf. Marcus [26]. 8 Knuth’s concept of treating a program as a piece of literature, addressed to human beings rather than to a computer...

...lically assisted proofs, and in the construction of a flexible computer environment in which researchers and research students can undertake such research. That is, in doing experimental mathematics. =-=[6]-=- Experimental Mathematics publishes formal results inspired by experimentation, conjectures suggested by experiments, surveys of areas of mathematics from the experimental point of view, descriptions ...

...ving the typology of the finite simple groups required a total of about fifteen thousand pages, spread in five-hundred separate articles belonging to about three-hundred different authors (see Conder =-=[15]-=-). But Serre [31] is still waiting for experts to check the claim by Aschbacher and Smith to have succeeded filling in the gap in the proof of the classification theorem, a gap already discovered in 1...

...ncy of rigor was given recently by one of the most prestigious mathematical journals, situated for a long time at the top of mathematical creativity: Annals of Mathematics. We learn from Karl Sigmund =-=[35]-=- that the proof proposed by Thomas Hales in August 1998 and the corresponding joint paper by Hales and Ferguson confirming Kepler’s conjecture about the densest possible packing of unit spheres into a...

... result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. This is a linear-growth model of mathematics (see Stöltzner =-=[36]-=-), a process in two stages. First, informal ideas are guessed and developed, conjectures are made, and outlines of justifications are suggested. Secondly, conjectures and speculations are tested and c...

... We confine ourselves to a few examples only: ...if one can program a computer to perform some part of mathematics, then in a very effective sense one does understand that part of mathematics (G. Tee =-=[38]-=-) If I can give an abstract proof of something, I’m reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I’m much happier. I’m rather an addict of doing thin...